Connections within the group of rounding and congruence functions and with other function groups

The rounding and congruence functions , , , , , , and have numerous representations through related functions, which are shown in the following tables, where the symbol means the characteristic function of a set (having the value 1 when its argument is an element of the specified set 𝔸, and a value of 0 otherwise):

The rounding and congruence functions , , , , , and can also be represented through elementary functions by the following formulas: